We give a bordism-theoretic characterization of those closed almost contact (2q+1)-manifolds (with q≥2) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
Bowden, J., Crowley, D., & Stipsicz, A. I. (2014). The topology of Stein fillable manifolds in high dimensions I. Proceedings of the London Mathematical Society, 109(6), 1363-1401. https://doi.org/10.1112/plms/pdu028